FORCING FOR hL AND hd

The present paper addresses the problem of attainment of the supremums in various equivalent definitions of hereditary density hd and hereditary Lindelöf degree hL of Boolean algebras. We partially answer two problems of J. Donald Monk, [13, Problems 50, 54], showing consistency of different attainment behaviour and proving that (for the considered variants) this is the best result we can expect. We deal with the attainment problem in various definitions of two cardinal functions on Boolean algebras: the hereditary density hd and the hereditary Lindelöf degree hL. These two cardinal functions are closely related, as it is transparent when we pick the right variants of (equivalent) definitions. Also they both are somewhat related to the spread s of Boolean algebras. So, for a Boolean algebra B, we define • s(B) = sup{κ : there is an ideal–independent sequence of length κ }, • hd(B) = sup{κ : there is a left–separated sequence of length κ }, • hL(B) = sup{κ : there is a right–separated sequence of length κ }. Let us recall that a sequence a ξ : ξ < κ of elements of a Boolean algebra is • ideal–independent if a ξ ζ∈w a ζ for each ξ < κ and a finite set w ⊆ κ \ {ξ}, • left–separated if a ξ ζ∈w a ζ for each ξ < κ and a finite set w ⊆ κ \ (ξ + 1), • right–separated if a ξ ζ∈w a ζ for each ξ < κ and a finite set w ⊆ ξ. The above definitions of the three cardinal functions are of special use, see e.g. [15, §1]. However, neither these definitions explain the names of the functions, nor they are good enough justifications for the interest in them. But all three functions originate in the cardinal functions of the topological space Ult(B) (of ultrafilters on B). And thus, for a Boolean algebra B, we may define (or prove that the following equalities hold true): • s(B) = sup{|X| : X ⊆ Ult(B) is discrete in the relative topology }, • hd(B) = sup{L(X) : X ⊆ Ult(B)}, where L(X) = min{κ : every open cover of X has a subcover of size ≤ κ }. The respective pairs of cardinal numbers are defined using sup, so even if we know that they are equal we still may expect different attainment properties: one of the families of …